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Theorem bcthlem2 23922

Description: Lemma for bcth 23926. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

**Assertion**
Ref	Expression
bcthlem2	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Distinct variable groups: 𝑘,𝑛,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝑔,𝑘,𝑛,𝑟,𝑥,𝑧,𝐷 𝑔,𝐹,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝐽,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝑀,𝑘,𝑛,𝑟,𝑥,𝑧 𝜑,𝑘,𝑛,𝑟,𝑥,𝑧 𝑥,𝑅 𝑔,𝑋,𝑘,𝑛,𝑟,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑧,𝑔,𝑘,𝑛) 𝑅(𝑧,𝑔,𝑘,𝑛,𝑟)

**Proof of Theorem bcthlem2**
Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
2		fvoveq1 7173	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1)))
3		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
4		fveq2 6664	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
5	3, 4	oveq12d 7168	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛)))
6	2, 5	eleq12d 2907	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
7	6	rspccva 3621	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
8	1, 7	sylan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
9		bcthlem.9	. . . . . 6 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
10	9	ffvelrnda 6845	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))
11		bcth.2	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
12		bcthlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
13		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
14	11, 12, 13	bcthlem1 23921	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
15	14	expr 459	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ⁺) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))))
16	10, 15	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
17	8, 16	mpbid 234	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))
18		cmetmet 23883	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
19	12, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
20		metxmet 22938	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
22	11	mopntop 23044	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
24		xp1st 7715	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋)
25		xp2nd 7716	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ⁺)
26	25	rpxrd 12426	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)
27	24, 26	jca 514	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*))
28		blssm 23022	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
29	28	3expb 1116	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
30	21, 27, 29	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
31		1st2nd2 7722	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (𝑔‘(𝑛 + 1)) = ⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
32	31	fveq2d 6668	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩))
33		df-ov 7153	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
34	32, 33	syl6reqr 2875	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
35	34	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
36	11	mopnuni 23045	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
38	37	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → 𝑋 = ∪ 𝐽)
39	30, 35, 38	3sstr3d 4012	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽)
40		eqid 2821	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	sscls 21658	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
42	23, 39, 41	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
43		difss2 4109	. . . . . . . 8 ⊢ (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
44		sstr2 3973	. . . . . . . 8 ⊢ (((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
45	42, 43, 44	syl2im 40	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
46	45	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))
47	46	ex 415	. . . . 5 ⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))))
48	47	3impd 1344	. . . 4 ⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
49	48	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
50	17, 49	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
51	50	ralrimiva 3182	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 ⊆ wss 3935 ⟨cop 4566 ∪ cuni 4831 class class class wbr 5058 {copab 5120 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1^st c1st 7681 2^nd c2nd 7682 1c1 10532 + caddc 10534 ℝ^*cxr 10668 < clt 10669 / cdiv 11291 ℕcn 11632 ℝ⁺crp 12383 ∞Metcxmet 20524 Metcmet 20525 ballcbl 20526 MetOpencmopn 20529 Topctop 21495 Clsdccld 21618 clsccl 21620 CMetccmet 23851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-cls 21623 df-cmet 23854

This theorem is referenced by: bcthlem3 23923 bcthlem4 23924

W3C validator